Cayley-Hamilton Theorem - Mathematical Methods of Physics, UGC - NET Physics | Physics for IIT JAM, UGC - NET, CSIR NET PDF Download

Introduction

• Cayley-Hamilton Theorem is quite an important theorem used in matrix theory.
• The Cayley Hamilton Theorem states that all real and complex square matrices will satisfy their own characteristic polynomial equation. This implies that when a square matrix is transformed into a polynomial, then this polynomial will be equal to 0.
• It is a very important result that is used in advanced linear algebra to simplify linear transformations.

What is Cayley Hamilton Theorem?

• According to the Cayley-Hamilton theorem, a square matrix will satisfy its own characteristic polynomial equation. 
• A characteristic polynomial is associated with the determinant of a matrix and the eigenvalues of the matrix will be the roots of this polynomial. 
• Suppose a square matrix A is given with n rows and n columns. 
• The characteristic polynomial of this matrix is given as det(λIₙ – A). 
  Here, Iₙ is the identity matrix, λ is a scalar quantity and det signifies the determinant operation.

Cayley Hamilton Theorem Statement

Cayley Hamilton Theorem Statement

p(A) = Aⁿ + aₙ₋₁Aⁿ⁻¹ + ... + a₁A + a₀Iₙ = 0
(OR)
p(A) = 0, where A is an n x n square matrix

• The Cayley-Hamilton theorem states that the characteristic polynomial expression of a real or complex square matrix will be equal to the zero matrix. 
• The characteristic polynomial p(λ) = det(λIₙ – A) can be decomposed as p(λ) = aₙλⁿ + aₙ₋₁λⁿ⁻¹ + ... + a₁λ + a₀. 
  This is a monic polynomial where the leading coefficient, i.e., the coefficient of the highest degree variable, will be equal to 1. 
  Thus, aₙ = 1. Here, aₙ₋₁, ..., a₁, a₀ are coefficients of the variables λⁿ⁻¹, ..., λ¹, λ⁰ respectively.

We have:
p(λ) = aₙλⁿ + aₙ₋₁λⁿ⁻¹ + ... + a₁λ + a₀
p(λ) = λⁿ + aₙ₋₁λⁿ⁻¹ + ... + a₁λ + a₀

On replacing λ with the matrix A, the polynomial can be written as follows:

p(A) = Aⁿ + aₙ₋₁Aⁿ⁻¹ + ... + a₁A + a₀Iₙ

Now according to the Cayley Hamilton Theorem, this polynomial will be 0.
Thus, p(A) = Aⁿ + aₙ₋₁Aⁿ⁻¹ + ... + a₁A + a₀Iₙ = 0 or p(A) = 0

Cayley Hamilton Theorem Formula

• The Cayley Hamilton theorem formula is extremely useful in performing complicated calculations with speed and accuracy. It can also be used to determine the inverse of a matrix. The formula is given as follows:
• Suppose the characteristic polynomial of an n × n square matrix, A, is given as:
  p(λ) = λⁿ + aₙ₋₁λⁿ⁻¹ + ... + a₁λ + a₀
  Then,
  p(A) = Aⁿ + aₙ₋₁Aⁿ⁻¹ + ... + a₁A + a₀Iₙ = 0
  Thus, p(A) = 0.
  To determine the inverse, multiply this equation with A⁻¹:
  Aⁿ⁻¹ + aₙ₋₁Aⁿ⁻² + ... + a₁Iₙ + a₀A⁻¹ = 0
  A⁻¹ = - (Aⁿ⁻¹ + aₙ₋₁Aⁿ⁻² + ... + a₁Iₙ) / a₀

Solved Examples

The examples based on Cayley Hamilton theorem are illustrated below:
Example 1 :  Prove Cayley Hamilton theorem for the following matrix?

Sol:  Let A =

Let us find characteristic polynomial of given matrix.

In order to prove the statement of Cayley Hamilton theorem for A, we need to show that:

P(A) = O, hence Cayley Hamilton theorem for given matrix A is proved.

Example 2 : If Cayley Hamilton theorem holds for the matrix, then find its inverse.

Its characteristic polynomial is -

According to Cayley Hamilton theorem -

Therefore, equation (1) becomes - 

 

Applications

• Cayley Hamilton theorem is widely applicable in many fields not only related to mathematics, but in other scientific fields too. 
• This theorem is used all over in linear algebra. 
• One can easily find inverse of a matrix using Cayley Hamilton theorem. 
• It also plays an important role in solving ordinary differential equations. 
• This theorem is quite useful in physics also. Cayley Hamilton theorem plays a vital role in computer programming and coding. 
  In a newer subject - Rheology, where behaviour of material is studied, this theorem is used to determine the equations that illustrate nature of materials.